What are the divisors of 1967?

1, 7, 281, 1967

There is a total of 4 positive divisors.
The sum of these divisors is 2256.
The arithmetic mean is 564.

4 odd divisors

1, 7, 281, 1967

How to compute the divisors of 1967?

A number N is said to be divisible by a number M (with M non-zero) if, when we divide N by M, the remainder of the division is zero.

$N mod M = 0$

Brute force algorithm

We could start by using a brute-force method which would involve dividing 1967 by each of the numbers from 1 to 1967 to determine which ones have a remainder equal to 0.

$Remainder = N - (M \times ⌊ \frac{N}{M} ⌋)$

(where $⌊ \frac{N}{M} ⌋$ is the integer part of the quotient)

1967 / 1 = 1967 (the remainder is 0, so 1 is a divisor of 1967)
1967 / 2 = 983.5 (the remainder is 1, so 2 is not a divisor of 1967)
1967 / 3 = 655.66666666667 (the remainder is 2, so 3 is not a divisor of 1967)
...
1967 / 1966 = 1.000508646999 (the remainder is 1, so 1966 is not a divisor of 1967)
1967 / 1967 = 1 (the remainder is 0, so 1967 is a divisor of 1967)

Improved algorithm using square-root

However, there is another slightly better approach that reduces the number of iterations by testing only integers less than or equal to the square root of 1967 (i.e. 44.350873723073). Indeed, if a number N has a divisor D greater than its square root, then there is necessarily a smaller divisor d such that:

$D \times d = N$

(thus, if $\frac{N}{D} = d$ , then $\frac{N}{d} = D$ )

1967 / 1 = 1967 (the remainder is 0, so 1 and 1967 are divisors of 1967)
1967 / 2 = 983.5 (the remainder is 1, so 2 is not a divisor of 1967)
1967 / 3 = 655.66666666667 (the remainder is 2, so 3 is not a divisor of 1967)
...
1967 / 43 = 45.744186046512 (the remainder is 32, so 43 is not a divisor of 1967)
1967 / 44 = 44.704545454545 (the remainder is 31, so 44 is not a divisor of 1967)